Peugeot Pulsion starts only when you shake it

Nice morning, I am currently having big problems starting my scooter Peugeot Pulsion 125 ccm, first registration 04/20, 30,000 km, electronic key (funk). The problem is that it’s been a gamble for some time, whether the vehicle starts or not. Sometimes it’s completely easy. Then again, even immediately after a ride, it doesn’t start at all. In these cases, nothing happens at all: if you press the start button and turn to the right, the system drives though high, everything lights up, the ads update, but when starting, nothing happens at all, not even groaning engine turns like if the battery is too weak. In order to make it run, you have to shake strongly during the start. In these cases it starts sooner or later, although sometimes as hesitant as you know it with a weak battery. However, this is 100pro in order, even freshly charged and shows at the start constantly approx. 14V. The first attempt, without shaking, is often the first attempt at the start, but only two hours later it is not at all or only to jump with shaking. I am now at a loss and unfortunately the Peugeot specialist workshop has not succeeded in locating the problem for the time being. The next guess would be a connection with the rear light or to look at the wire harness?! Something even more complementary: as soon as the scooter drives, there are no abnormalities, what e.g. V use, speed, noise level etc. Maybe someone had a similar problem with the pulsion and can help me otherwise. Thank you!